Some Extensions and Analysis of Flux and Stress Theory Reuven Segev Department of Mechanical Engineering Ben-Gurion University Structures of the Mechanics of Complex Bodies October 2007 Centro di Ricerca Matematica, Ennio De Giorgi Scuola Normale Superiore R. Segev ( Ben-Gurion Univ. ) 1 / 20 Flux and Stress Theories Pisa, Oct. 2007
The Global Point of View C n -Functionals R. Segev ( Ben-Gurion Univ. ) 2 / 20 Flux and Stress Theories Pisa, Oct. 2007
Review of Basic Kinematics and Statics on Manifolds is con- T κ Q The mechanical system is man- characterized by its κ configuration space—a manifold Q . ec- Velocities are tangent vectors to the manifold—elements of T Q . a- A Force at the configuration κ is a linear mapping mapping Q F : T κ Q → R . Can we apply this framework to Continuum Mechanics? R. Segev ( Ben-Gurion Univ. ) 3 / 20 Flux and Stress Theories Pisa, Oct. 2007
Problems Associated with the Configuration Space in Continuum Mechanics What is a configuration? Does the configuration space have a structure of a manifold? The configuration space for continuum mechanics is infinite dimensional. R. Segev ( Ben-Gurion Univ. ) 4 / 20 Flux and Stress Theories Pisa, Oct. 2007
Configurations of Bodies in Space A mapping of the body into space; material impenetrability—one-to-one; not “crash” volumes—invertible derivative. continuous deformation gradient (derivative); do not “crash” volumes—invertible derivative. κ κ ( B ) A body B Space ✪ U R. Segev ( Ben-Gurion Univ. ) 5 / 20 Flux and Stress Theories Pisa, Oct. 2007
Manifold Structure for Euclidean Geometry If the body is a subset of R 3 and space is modeled by R 3 , the collection of differentiable mappings C 1 ( B , R 3 ) is a vector space However, the subset of “good” configurations is not a vector space, e.g., κ − κ = 0 —not one-to-one. We want to make sure that the subset of configurations Q is an open subset of C 1 ( B , R 3 ) , so it is a trivial manifold. configurations configurations C 1 ( B , R 3 ) C 1 ( B , R 3 ) all di c erentiable mappings all di c erentiable mappings R. Segev ( Ben-Gurion Univ. ) 6 / 20 Flux and Stress Theories Pisa, Oct. 2007
The C 0 -Distance Between Functions The C 0 -distance between functions measures the maximum difference between functions. A configuration is arbitrarily close to a “bad” mapping. Space a configuration solid “bad mapping” dotted Body R. Segev ( Ben-Gurion Univ. ) 7 / 20 Flux and Stress Theories Pisa, Oct. 2007
The C 1 -Distance Between Functions The C 1 distance between functions measures the maximum difference between functions and their derivative | u − v | C 1 = sup {| u ( x ) − v ( x ) | , | Du ( x ) − Dv ( x ) |} . A configuration is always a finite distance away from a “bad” mapping. Space a configuration solid “bad mapping” dotted Body R. Segev ( Ben-Gurion Univ. ) 8 / 20 Flux and Stress Theories Pisa, Oct. 2007
Conclusions for R 3 If we use the C 1 -norm, the configuration space of a continuous body in space is an open subset of C 1 ( B , R 3 ) -the vector space of all u ( κ ( x )) = d κ ( x ) differentiable mapping. dt Q is a trivial infinite dimensional κ { B } manifold and its tangent space at any point may be identified with C 1 ( B , R 3 ) . A tangent vector is a velocity field. R. Segev ( Ben-Gurion Univ. ) 9 / 20 Flux and Stress Theories Pisa, Oct. 2007
For Manifolds Both the body B and space U are differentiable manifolds. The configuration space is the collection Q = Emb ( B , U ) of the embeddings of the body in space. This is an open submanifold of the infinite dimensional manifold C 1 ( B , U ) . The tangent space T κ Q may be characterized as ∗ → | ◦ } T κ Q = C 1 ( κ ∗ T U ) . T κ Q = { w : B → T Q | τ ◦ w = κ } , or alternatively, κ T M κ ∗ ( T M ) T x M projection τ w x B a body κ M space manifold R. Segev ( Ben-Gurion Univ. ) 10 / 20 Flux and Stress Theories Pisa, Oct. 2007
Representation of C 0 -Functionals by Integrals Assume you measure the size of a function using the C 0 -distance, � w � = sup {| w ( x ) |} . A linear functional F : w �→ F ( w ) is continuous with respect to this norm if F ( w ) → 0 when max | w ( x ) | → 0 . Riesz representation theorem: A continuous linear functional F with respect to the C 0 -norm may be represented by a unique measure µ in the form � F ( w ) = w d µ . B F ( w ) = � F ( w ) = � w φ dx w φ dx B B δ δ F isn’t sensitive to the derivative force density force density φ φ Velocity Velocity w w Body B Body B R. Segev ( Ben-Gurion Univ. ) 11 / 20 Flux and Stress Theories Pisa, Oct. 2007
Representation of C 1 -Functionals by Integrals Now, you measure the size of a function using the C 1 -distance, � w � = sup {| w ( x ) | , | Dw ( x ) |} . A linear functional F : w �→ F ( w ) is continuous with respect to this norm if F ( w ) → 0 when both max | w ( x ) | → 0 and max | Dw ( x ) | → 0 . Representation theorem: A continuous linear functional F with respect to the C 1 -norm may be represented by measures σ 0 , σ 1 in the form � � F ( w ) = w d σ 0 + Dw d σ 1 . B B F is sensitive to the derivative F ( w ) = � φ 0 wdx + � φ 1 Dwdx B B δ “self” force density φ 0 stress density φ 1 Velocity w Body B ✪ Velocity gradient Dw R. Segev ( Ben-Gurion Univ. ) 12 / 20 Flux and Stress Theories Pisa, Oct. 2007
Non-Uniqueness of C 1 -Representation by Integrals We had an expression in the form � � w ′ d σ 1 . F ( w ) = w d σ 0 + B B If we were allowed to vary w and w ′ independently, we could determine σ 0 and σ 1 uniquely. This cannot be done because of the condition w ′ = Dw . • F ( w ) = � φ 0 wdx + � φ 1 w ′ dx B B δ “self” force density φ 0 stress density φ 1 Velocity w Body B w ′ ✪ R. Segev ( Ben-Gurion Univ. ) 13 / 20 Flux and Stress Theories Pisa, Oct. 2007
Unique Representation of a Force System Assume we have a force system, i.e., a force F P for every subbody P of B . We can approximate pairs of non-compatible functions w and w ′ , i.e., w ′ � = Dw , by piecewise compatible functions. w ′ � Dw approximation of � w B wd σ 0 Calculate � B wd σ 0 Body B P 1 P 2 . . . . . . B w ′ d σ 1 approximation of � w ′ B w ′ d σ 1 � Calculate P 1 P 2 Body B . . . . . . This way the two measures are determined uniquely. One needs consistency conditions for the force system. R. Segev ( Ben-Gurion Univ. ) 14 / 20 Flux and Stress Theories Pisa, Oct. 2007
Generalized Cauchy Consistency Conditions P 1 • Additivity: P 2 F P 1 ∪ P 2 ( w | P 1 ∪ P 2 ) = F P 1 ( w | P 1 ) + F P 2 ( w | P 2 ) . • Continuity: If P i → A , then F P i ( w | P 1 ) P i converges and the limit depends on A only. A • Uniform Boundedness: There is a K > 0 such that for every subbody P and every w , | F P ( w | P ) ≤ K � w P � . Main Tool in Proof: Approximation of measurable sets by bodies with smooth boundaries. R. Segev ( Ben-Gurion Univ. ) 15 / 20 Flux and Stress Theories Pisa, Oct. 2007
Generalizations All the above may be formulated and proved for differentiable manifolds. This formulation applies to continuum mechanics of order k > 1 (stress tensors of order k ). One should simply use the C k -norm instead of the C 1 -norm. The generalized Cauchy conditions also apply to continuum mechanics of order k > 1 . This is the only formulation of Cauchy conditions for higher order continuum mechanics. R. Segev ( Ben-Gurion Univ. ) 16 / 20 Flux and Stress Theories Pisa, Oct. 2007
Locality and Continuity in Constitutive Theory R. Segev ( Ben-Gurion Univ. ) 17 / 20 Flux and Stress Theories Pisa, Oct. 2007
Global Constitutive Relations (Elasticity for Simplicity) Q , the configuration space of a body B . C 0 � � R 3 , R 3 �� B , L , the collection of all stress fields over the body. Ψ : Q → C 0 � � R 3 , R 3 �� B , L , a global constitutive relation. � � �� space stress Ψ configuration stress field κ σ = Ψ ( κ ) Global constitutive relation. Body B Body B R. Segev ( Ben-Gurion Univ. ) 18 / 20 Flux and Stress Theories Pisa, Oct. 2007
Locality and Materials of Grade- n Germ Locality: If two configurations κ 1 and κ 2 are equal on a subbody containing X , then the resulting stress fields are equal at X . space stress Ψ Ψ ( κ 1 ) κ 1 Ψ ( κ 2 ) κ 2 X X P P Body B Body B Material of Grade-n or n-Jet Locality: If the first n derivatives n n n of κ 1 and κ 2 are equal at X , then, Ψ ( κ 1 )( X ) = Ψ ( κ 2 )( X ) . (Elastic = grade 1.) X X X 1 2 space stress Ψ Ψ ( κ 1 ) κ 1 Ψ ( κ 2 ) κ 2 X X P P Body B Body B R. Segev ( Ben-Gurion Univ. ) 19 / 20 Flux and Stress Theories Pisa, Oct. 2007
n -Jet Locality and Continuity Basic Theorem: If a constitutive relation Ψ : Q → C 0 � � R 3 , R 3 �� B , L is local and continuous with respect to the C n -norm, then, it is n -jet local. In particular, if Ψ is continuous with respect to the C 1 -topology, the material is elastic. space space space Whitney’s restriction extension κ ′ κ 1 | P 1 κ 1 κ ′ κ 2 | P 2 κ 2 X X X P P Body B Body B Body B Ψ Ψ stress stress Ψ ( κ ′ Ψ ( κ 1 ) 1 ) Ψ ( κ ′ Ψ ( κ 2 ) 2 ) X X P P Body B Body B R. Segev ( Ben-Gurion Univ. ) 20 / 20 Flux and Stress Theories Pisa, Oct. 2007
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